Found problems: 4776
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:
$\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$
A function $f : R -\{0\} \to R$ has the following properties:
(i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$,
(ii) $f$ takes the value $\frac12$ at least once.
Determine $f(-1)$.
Prove that $f$ is a periodic function
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.
The function $f(x)$ has the property that, for some real positive constant $C$, the expression \[\frac{f^{(n)}(x)}{n+x+C}\] is independent of $n$ for all nonnegative integers $n$, provided that $n+x+C\neq 0$. Given that $f'(0)=1$ and $\int_{0}^{1}f(x) \, dx = C+(e-2)$, determine the value of $C$.
Note: $f^{(n)}(x)$ is the $n$-th derivative of $f(x)$, and $f^{(0)}(x)$ is defined to be $f(x)$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:
$(i)$ $f(x)=-f(-x);$
$(ii)$ $f(x+1)=f(x)+1;$
$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
Given the function $$y =|x^2 - 4x + 3|.$$
Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy
\[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$
[list=a]
[*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$
[*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$
for any $x,y$ positive real numbers.
suppose that $S_{\mathbb N}$ is the set of all permutations of natural numbers. finite permutations are a subset of $S_{\mathbb N}$ that behave like the identity permutation from somewhere. in other words bijective functions like $\pi: \mathbb N \longrightarrow \mathbb N$ that only for finite natural numbers $i$, $\pi(i)\neq i$. prove that we cannot put probability measure that is countably additive on $\wp(S_{\mathbb N})$ (family of all the subsets of $S_{\mathbb N}$) that is invarient under finite permutations.
Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times).
a) Find the number of distinct real roots of the equation $f^{3}(x) = x$
b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:
$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$
for all $x,y \in \mathbb{R}$
[i]Proposed by Nikolay Nikolov[/i]
Let $\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\inf_{P\in\mathcal{P}_n}\int_a^b |P(x)|\, dx >0.$$
[i]Cristi Săvescu[/i]
Let $f : [a, b] \rightarrow [a, b]$ be a continuous function and let $p \in [a, b]$. Define $p_0 = p$ and $p_{n+1} = f(p_n)$ for $n = 0, 1, 2,...$. Suppose that the set $T_p = \{p_n : n = 0, 1, 2,...\}$ is closed, i.e., if $x \not\in T_p$ then $\exists \delta > 0$ such that for all $x' \in T_p$ we have $|x'-x|\ge\delta$.
Show that $T_p$ has finitely many elements.
Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$
for all $x,y\in Q^+ .$
Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.
Let $ H$ be a set of real numbers that does not consist of $ 0$ alone and is closed under addition. Further, let $ f(x)$ be a
real-valued function defined on $ H$ and satisfying the following conditions: \[ \;f(x)\leq f(y)\ \mathrm{if} \;x \leq y\] and \[ f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ .\] Prove that $ f(x)\equal{}cx$ on $ H$, where $ c$ is a nonnegative number. [M. Hosszu, R. Borges]
Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded?
[b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.
$ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $
Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f'(x)=\frac{f(x+n)-f(x)}n\]
for all real numbers $x$ and all positive integers $n.$
Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$
Prove that for every $ \alpha\in(0,1),$
\[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]